#8186968

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#59675	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,4)(5,8)(6,14)(7,10)(9,15)(11,13)(12,16)(17,18)(19,25)(20,21)(22,26)(23,24)(27,32)(28,29)(30,31)(33,34,38,42)(35,43,45,39)(36,46,44,40)(37,41,47,48);(1,3,7,8)(2,4,10,5)(6,11,15,16)(9,12,14,13)(18,24)(20,25)(26,31)(28,32)(33,35,38,45)(34,39,42,43)(36,37,44,47)(40,41,46,48);(1,4,7,5)(2,8,10,3)(6,12,15,13)(9,16,14,11)(17,19,23,21)(18,25,24,20)(22,29,30,27)(26,28,31,32)(33,36)(34,40)(35,37)(38,44)(39,41)(42,46)(43,48)(45,47);(1,5,7,4)(2,8,10,3)(6,13,15,12)(9,16,14,11)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,31)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,46)(47,48);(1,6,7,15)(2,9,10,14)(3,11,8,16)(4,12,5,13)(17,21,23,19)(18,20,24,25)(22,29,30,27)(26,28,31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,46)(47,48);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,16)(14,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,37)(34,41)(35,44)(36,45)(38,47)(39,46)(40,43)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094164);